Equivalently, a group is complete if the conjugation map, G → Aut(G) (sending an element g to conjugation by g), is an isomorphism: injectivity implies the group is centerless, as no inner automorphisms are the identity, while surjectivity implies it has no outer automorphisms.
A complete group is always isomorphic to its automorphism group (via sending an element to conjugation by that element), although the reverse need not hold: for example, the dihedral group of 8 elements is isomorphic to its automorphism group, but it is not complete. For a discussion, see (Robinson 1996, section 13.5).
Extensions of complete groups
Assume that a group, G, is a group extension given as a short exact sequence of groups
- 1 ⟶ N ⟶ G ⟶ G′ ⟶ 1
with kernel, N, and quotient, G′. If the kernel, N, is a complete group then the extension splits: G is isomorphic to the direct product, N × G′. A proof using homomorphisms and exact sequences can be given in a natural way: The action of G (by conjugation) on the normal subgroup, N gives rise to a group homomorphism, φ: G → Aut(N) ≅ N. Since Out(N) = 1 and N has trivial center the homomorphism φ is surjective and has an obvious section given by the inclusion of N in G. The kernel of φ is the centralizer CG(N) of N in G, and so G is at least a semidirect product, CG(N) ⋊ N, but the action of N on CG(N) is trivial, and so the product is direct. This proof is somewhat interesting since the original exact sequence is reversed during the proof.
This can be restated in terms of elements and internal conditions: If N is a normal, complete subgroup of a group, G, then G = CG(N) × N is a direct product. The proof follows directly from the definition: N is centerless giving CG(N) ∩ N is trivial. If g is an element of G then it induces an automorphism of N by conjugation, but N = Aut(N) and this conjugation must be equal to conjugation by some element n of N. Then conjugation by gn−1 is the identity on N and so gn−1 is in CG(N) and every element, g, of G is a product (gn−1)n in CG(N)N.
- Robinson, Derek John Scott (1996), A course in the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6
- Rotman, Joseph J. (1994), An introduction to the theory of groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94285-8 (chapter 7, in particular theorems 7.15 and 7.17).